Question 709320: A graphing iron is thrown vertically to catch on a ledge 7.5 metres above the thrower. It`s height is give in meters and time in seconds. After being thrown the equation is given by h= -4.9t?2+11t+1.5. Will the graphing iron reach the ledge?
Can you please help me? Thank you so much in advance:)
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! I hope that when you look at the equation:
that you would recognize that the graph of this equation would be a parabola. And because of the negative, -4.9, in front the  , this parabola will open downward.
If you picture such a parabola, I hope you will realize that the vertex of the parabola will be the highest point. The highest point will have the largest h coordinate. In other words, the vertex will tell us the highest height the "graphing iron" (actually I think the word is grappling) will reach. If this greatest height of the iron is less than the height of the ledge then the iron will not reach the ledge. If the greatest height of the iron is greater than or equal to the height of the ledge then the iron will reach the ledge.
So if we find the vertex and then we will be able to answer the question. One way to find the vertex is to transform the equation into vertex form:  . But this way requires that we complete the square and do some factoring. Another way to find the vertex is from knowing that the x (or t) coordinate of the vertex of a parabola whose equation is in standard form,  or in our case  , will always be -b/2a. This is how I will proceed. The "a" of our equation is -4.9 and the "b" is 11. So the x (or t) coordinate of the vertex will be:
which simplifies to:
We can eliminate the decimal by multiplying the numerator and denominator by 10:
which reduces to:
This is the t value, the time, of the vertex. It is not the maximum height. The 55/29 tells us that the iron will reach its highest point after 55/49 seconds. We need to find the y (of h) coordinate of the vertex. For this we will substitute in the t value of the vertex into our equation:
Since close is probably good enough, I am going to be lazy and use a calculator and round off the decimals (the 55/49 will be an infinite, repeating decimal).
So the maximum height will be approximately 7.65 meters. So the iron will reach a 7.5 meter ledge.
Note: The rounding done with the decimals probably would not make the actual, exact maximum height be lower than 7.5. But if you want to be sure (or if your teacher doesn't want you to use decimal approximations) then you should rework the calculations for h using fractions to get the exact maximum height. I'll get you started:
If you keep everything as fractions as you work this out you will get an exact value for h.
In case you're curious, here is the graph of the parabola:
The horizontal line near the vertex is where h = 7.5. So you can see that the graph seems to agree that the iron will reach the ledge.
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